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SVD -- singular value decomposition of a rectangular matrix over RR or CC

Synopsis

Usage:
(S,U,Vt) = SVD M
Inputs:
- M, a matrix, over RR or CC, of size m by n
Outputs:
- S, a matrix, over RR, the singular values, of size t by 1, where t = min(m,n)
- U, a matrix, an orthogonal (unitary) matrix of size m by m
- Vt, a matrix, an orthogonal (unitary) matrix of size n by n
Optional inputs:
- DivideConquer => ..., -- Use the lapack divide and conquer SVD algorithm

Description

If Sigma is the m by n matrix whose (i,i) entry is the i th entry of S, and zero elsewhere, then M = U Sigma Vt. This is the singular value decomposition of M. Note that the entries of S are (up to roundoff error) the eigenvalues of the Hermitian matrix M * (conjugate transpose M)

M may also be a MutableMatrix in which case the returned values S, U, Vt are also mutable matrices.

If M is over CC, then U and Vt are unitary matrices over CC. If M is over RR, U and Vt are orthogonal over RR.

i1 : M = map(RR^3, RR^5, (i,j) -> (i+1)^j * 1.0)

o1 = | 1 1        1        1         1         |
     | 1 2.000000 4.000000 8.000000  16.000000 |
     | 1 3.000000 9.000000 27.000000 81.000000 |

              3        5
o1 : Matrix RR  <--- RR

i2 : (S,U,V) = SVD(M)

o2 = (| 87.813396 |, | -0.016192 -0.401463 -0.915732 |, | -0.013677 -0.038310
      | 3.895099  |  | -0.206478 -0.894769 0.395923  |  | -0.282597 -0.411936
      | 0.797285  |  | -0.978317 0.195490  -0.068405 |  | -0.737771 -0.412778
                                                        | 0.083862  0.332631 
                                                        | -0.607135 0.740143 
     ------------------------------------------------------------------------
     -0.109857 -0.319798 -0.940216 |)
     -0.570238 -0.585709 0.286743  |
     0.065613  0.507606  -0.152769 |
     -0.807831 0.472401  -0.081063 |
     0.076583  -0.271748 0.062156  |

o2 : Sequence

i3 : (transpose U) * M * (transpose V)

o3 = | 87.813396 0.000000  0.000000  0.000000 0.000000  |
     | -0.000000 3.895099  -0.000000 0.000000 -0.000000 |
     | -0.000000 -0.000000 0.797285  0.000000 0.000000  |

              3        5
o3 : Matrix RR  <--- RR

i4 : U^-1 == transpose U

o4 = true

i5 : (S1,U1,V1) = SVD(M, DivideConquer => true)

o5 = (| 87.813396 |, | -0.016192 -0.401463 -0.915732 |, | -0.013677 -0.038310
      | 3.895099  |  | -0.206478 -0.894769 0.395923  |  | -0.282597 -0.411936
      | 0.797285  |  | -0.978317 0.195490  -0.068405 |  | -0.737771 -0.412778
                                                        | 0.083862  0.332631 
                                                        | -0.607135 0.740143 
     ------------------------------------------------------------------------
     -0.109857 -0.319798 -0.940216 |)
     -0.570238 -0.585709 0.286743  |
     0.065613  0.507606  -0.152769 |
     -0.807831 0.472401  -0.081063 |
     0.076583  -0.271748 0.062156  |

o5 : Sequence

i6 : S1 == S, U1==U, V1==V

o6 = (true, true, true)

o6 : Sequence

The SVD routine calls on the SVD algorithms in the lapack library.

See also

eigenvalues -- find eigenvalues of a matrix over RR or CC
eigenvectors -- find eigenvectors of a matrix over RR or CC

Ways to use `SVD` :

SVD(Matrix)
SVD(MutableMatrix)